A Class of Fuzzy Variational Inequality Based on Monotonicity of Fuzzy Mappings

Definition 1 (see Wu and Xu [6].)A fuzzy number μ is a fuzzy set with the following properties:

• (1)

  μ  is normal; that is, there exists x₀ ∈ Rⁿ such that μ(x₀) = 1;

• (2)

  μ is convex fuzzy set; that is, μ(λx + (1 − λ)y) ≥ min (μ(x), μ(y)), x, y ∈ Rⁿ, λ ∈ [0,1];

• (3)

  [μ] ⁰ is compact.

Let E denote the family of fuzzy numbers; that is, E denotes the family of compact and convex fuzzy set on R¹. Obviously, [μ] ^α is a nonempty compact convex subset of R¹ (denoted by [μ_*(α), μ^*(α)]) for all μ ∈ E and for all α ∈ [0,1].

A precise number a is a special case of fuzzy number encoded as

However, a precise number will be denoted as usual, in particular, number 0. The fuzzy numbers μ, ν ∈ E are represented by (μ_*(α), μ^*(α)) and (ν_*(α), ν^*(α)), respectively. For each real number λ, the addition $$ and scalar multiplication λμ are defined as follows:

It is well known that for all μ, ν ∈ E and λ ∈ R¹

For x = (x₁, x₂, …, x_n), y = (y₁, y₂, …, y_n) ∈ Rⁿ, x ≤ y if and only if x_i ≤ y_i  (i = 1,2, …, n), and x < y if and only if x ≤ y and x ≠ y.

Definition 2. For μ, ν ∈ E, μ⪯ν, if and only if for every α ∈ [0,1], μ_*(α) ≤ ν_*(α) and μ^*(α) ≤ ν^*(α).

•   

  If μ⪯ν, ν⪯μ, then μ = ν.

•   

  μ≺ν if and only if μ⪯ν and ∃α₀ ∈ [0,1], such that μ_*(α₀) < ν_*(α₀) or ν_*(α₀) < μ_*(α₀).

For μ, ν ∈ E, if either μ⪯ν or ν⪯μ, then μ and ν are comparable; otherwise, they are noncomparable.

If μ, ν ∈ E, there exists ω ∈ E such that $$, then we say the Hukuhara difference of μ and ν exists, call ω the H-difference of μ and ν, and denote $$.

It is obvious that if the H-difference $$ exists, then $$, $$.

Definition 3. A mapping F : K(⊂Rⁿ) → E is said to be a fuzzy mapping. Denote [F(x)](α) = [F(x) _*(α), F(x) ^*(α)], for all α ∈ [0,1].

Definition 4 (see Buckley and Feuring [34].)Let F be a fuzzy mapping from the set of real numbers R to the set of all fuzzy numbers, and let [F(x)](α) = [F_*(x)(α), F^*(x)(α)]. Assume that the partial derivatives of F_*(x)(α), F^*(x)(α) with respect to x ∈ R for each α ∈ [0,1] exist and are denoted by $$, $$, respectively. Let $$ for x ∈ R, α ∈ [0,1]. If Γ(x, α) defines the α-cut of a fuzzy number for each x ∈ R, then F(x) is said to be differentiable and is written as $$, for all x ∈ R, α ∈ [0,1].

Definition 5 (see Panigrahi et al. [4].)Let F : K(⊂Rⁿ) → E be a fuzzy mapping, where K ⊂ Rⁿ is an open set. Let x = (x₁, x₂, …, x_n) ∈ K. Let $$,  (i = 1,2, …, n) stand for the “partial differentiation” with respect to the ith variable x_i. Assume that, for all α ∈ [0,1], F_*(x)(α), F^*(x)(α) have continuous partial derivatives so that $$, $$ are continuous. Define

If each i = 1,2, …, n, $$ defines the α-cut of a fuzzy number, then F is called differentiable at x, and it can be represented as

$$ is said to be the gradient of the fuzzy mapping F at x.

U is said to be an n-dimensional fuzzy vector if and only if the components of U are composed by n fuzzy numbers, denoted by U = (μ₁, μ₂, …, μ_n) ^T. The set of all n-dimensional fuzzy vectors is denoted by (E) ⁿ.

A λ level vector of fuzzy vector U = (μ₁, μ₂, …, μ_n) ^T is defined as

The addition and the scalar multiplication of fuzzy vectors U = (μ₁, μ₂, …, μ_n) ^T and V = (ν₁, ν₂, …, ν_n) ^T are defined as

Definition 6. A differentiable comparable fuzzy mapping F : K → E is said to be

• (a)

  fuzzy invex with respect to η : K × K → Rⁿ, if and only if

  $$ ()

• (b)

  fuzzy incave with respect to η : K × K → Rⁿ, if and only if

  $$ ()

• (c)

  fuzzy strictly invex with respect to η : K × K → Rⁿ, if and only if

  $$ ()

• (d)

  fuzzy strictly incave with respect to η : K × K → Rⁿ, if and only if

  $$ ()

• (e)

  fuzzy pseudo-invex with respect to η : K × K → Rⁿ, if and only if

  $$ ()

• (f)

  fuzzy strictly pseudo-invex with respect to η : K × K → Rⁿ, if and only if

  $$ ()

Example 7 (see Wu and Xu [6].)Let F : K(⊂R¹) → E represent the reproduction rate of some germ:

So, [F(x)] ^α = [αx², (2 − α)x²], α ∈ [0,1]. Then there is η(x, y) = x − y, such that F(x) is a fuzzy invex mapping, where K = (0, ∞), x, y ∈ K, x represents the predicted quantity, and t represents the actual reproduction quantity.

Example 8. Consider the fuzzy mapping [F(x)] ^α = [αx², (2 − α)x²], α ∈ [0,1]. Then, there is an η(x, y) = x⁵ − y⁵ such that F(x) is a fuzzy pseudoinvex mapping, where K = (0, ∞), x, y ∈ K.

Remark 9. For an invex fuzzy mapping, there must exist η : K × K → Rⁿ, such that

holds.

Proof. Since F is a comparable fuzzy mapping, then for all x, y ∈ Rⁿ, there is

or Without loss of generality, suppose that Thus, for all α ∈ [0,1],

If ∇F(y) = 0, for any η(x, y) : K × K → Rⁿ, the result holds.

If ∇F(y) ≠ 0, for given α ∈ [0,1].

(1) When F^*(x)(α) − F^*(y)(α) ≥ F_*(x)(α) − F_*(y)(α).

(i) If ∇F_*(y)(α) > 0, then take

Thus, that is, From ∇F^*(y)(α)≥∇F_*(y)(α) > 0, there is so, Hence, That is, From (22) and (26), it follows that

(ii) If ∇F_*(y)(α) < 0, then take

Thus, From ∇F_*(y)(α) < 0 and ∇F^*(y)(α)≥∇F_*(y)(α), there is so, Hence, That is, From (29) and (33), it follows that

(iii) If ∇F_*(y)(α) is indefinite, there is a vector C ∈ Rⁿ, such that

holds.

Take

Thus, That is, From ∇F_*(y)(α)  +  C ≤ 0 and ∇F^*(y)(α)≥∇F_*(y)(α), there is so, Hence, That is, From (38) and (42), it follows that

(2) When F^*(x)(α) − F^*(y)(α) ≤ F_*(x)(α) − F_*(y)(α).

(i) If ∇F_*(y)(α) > 0, then take

Thus, On the other hand, That is, From (45) and (47), it follows that

(ii) If ∇F_*(y)(α) < 0, then take

Thus, That is, On the other hand, that is, From (51) and (53), it follows that

(iii) If ∇F_*(y)(α) is indefinite, there is a vector C ∈ Rⁿ, such that

holds.

Take

Thus, That is, On the other hand, that is, From (58) and (60), it follows that This completes the proof.

Definition 10. A comparable fuzzy mapping F : K → (E) ⁿ is said to be

• (a)

  fuzzy invex monotone on K, if ∃η : K × K → Rⁿ such that for any x, y ∈ K,

  $$ ()

• (b)

  fuzzy pseudo-invex monotone on K, if ∃η : K × K → Rⁿ such that for any x, y ∈ K,

  $$ ()

• (c)

  fuzzy strictly invex monotone on K, if ∃η : K × K → Rⁿ such that for any x, y ∈ K, x ≠ y,

  $$ ()

• (d)

  fuzzy strictly pseudo-invex monotone if and only if ∃η : K × K → Rⁿ such that for any x, y ∈ K, x ≠ y,

  $$ ()

Definition 11. The function η : K × K → Rⁿ is said to be a skew function if

Definition 12. Let y ∈ K, K is said to be invex at y with respect to η : K × K → Rⁿ if, for each x ∈ K, λ ∈ [0,1],

K is said to be an invex set with respect to η if K is invex at each y ∈ K.

Theorem 13. If a differentiable fuzzy mapping F : K → E is invex on K with respect to η : K × K → Rⁿ and η is a skew function. Then, ∇F : K → (E) ⁿ is fuzzy invex monotone with respect to the same η.

Proof. Let F be invex on K, then there exists η(x, y) ∈ Rⁿ, such that

That is, there are for all α ∈ [0,1].

By changing x for y,

That is, there are for all α ∈ [0,1].

From (69) and (71), it follows that

As η(x, y) + η(y, x) = 0, then Therefore, from (73), there is

Corollary 14. If a differentiable fuzzy mapping F : K → E is invex on K with respect to η : K × K → Rⁿ and η is a skew function. Then, ∇F : K → (E) ⁿ is fuzzy pseudo-invex monotone with respect to the same η.

Proof. From Theorem 13, it follows that

Thus, for all α ∈ [0,1].

If η(y,x)^T∇F(x)⪰0,

for all α ∈ [0,1].

Thus, from (76) and (77),

holds. Therefore, η(y, x) ^T∇F(y)⪰0.

Theorem 15. If a differentiable fuzzy mapping F : K → E is strictly invex on K with respect to η : K × K → Rⁿ and η is a skew function. Then, ∇F : K → (E) ⁿ is fuzzy strictly invex monotone on K with respect to the same η.

Proof. Assume that F is strictly invex on K, then there exists η(x, y) ∈ Rⁿ, such that, for any x, y ∈ K, x ≠ y,

Thus, there exists some α₀ ∈ [0,1], such that or For α₀ ∈ [0,1], without loss of generality, suppose that By changing x for y, By (82), (83) and since η is a skew function, we have On the other hand, for other α ∈ [0,1], Therefore,

Theorem 16. If a differentiable fuzzy mapping F : K → E is strictly pseudo-invex on K with respect to η : K × K → Rⁿ and η is a skew function. Then, ∇F : K → (E) ⁿ is fuzzy strictly pseudo-invex monotone on K with respect to the same η.

Proof. Let F be a fuzzy strictly pseudo-invex, then there exists η(x, y) ∈ Rⁿ, such that for any x, y ∈ K, x ≠ y,

We need to show that there exists η(x, y) ∈ Rⁿ, such that for all x, y ∈ K, x ≠ y.

By contradiction, suppose that η(x, y) ^T∇F(x)⪯0, then there exists some α₀ ∈ [0,1], such that

or Without loss of generality, assume that As η(x, y) + η(y, x) = 0, then Since F is strictly pseudo-invex on K, then for α₀ ∈ [0,1], there is which is a contradiction.

Theorem 17. Let F : K → (E) ⁿ be a fuzzy strictly pseudo monotone mapping on K with respect to η : K × K → Rⁿ; then, F is a fuzzy pseudo monotone with respect to η on K.

Proof. As F is a strictly pseudo monotone with respect to η on K, then for any α ∈ [0,1],

F_*(x)(α), F^*(x)(α) is strictly pseudo monotone with respect to η on K. Thus, F_*(x)(α), F^*(x)(α) also is pseudo monotone with respect to η on K; that is, F is pseudo monotone with respect to η on K.

Theorem 18. Let F : K → E be a differentiable mapping, and suppose that

• (i)

  η satisfies the following conditions:

  • (a)

    η(y, y + tη(x, y)) = −tη(x, y),

  • (b)

    η(x, y + tη(x, y)) = (1 − t)η(x, y);

• (ii)

  K is an invex set with respect to η;

• (iii)

  for each x ≠ y, some α₀ ∈ [0,1],

  • (a)

    F_*(y)(α₀) > F_*(x)(α₀) implies $$,

•  

  or

  • (b)

    F^*(y)(α₀) > F^*(x)(α₀) implies $$;

• (iv)

  ∇F : K → (E) ⁿ is fuzzy pseudo-invex monotone with respect to η on K.

Then, F is a fuzzy pseudo-invex mapping η on K.

Proof. Set x, y ∈ K, x ≠ y, and η(x, y) ^T∇F(y)⪰0 holds. Thus, we need to show that F(x)⪰F(y); that is, F_*(x)(α) ≥ F_*(y)(α) and F^*(x)(α) ≥ F^*(y)(α), for all α ∈ [0,1].

Assume the contrary, that is, F(x)F(y). Thus, there exists some α₀ ∈ [0,1], such that

or Without loss of generality, suppose that By hypothesis (iii), for some $$.

It follows from (98) and (i) that

for some $$.

Since ∇F is a pseudo-invex monotone with respect to η, thus

From $$ and $$, (100) becomes This contradicts η(x, y) ^T∇F^*(y)(α)⪰0.

Theorem 19. Let F : K → E be a differentiable mapping, and suppose that

• (i)

  η satisfies the following conditions:

  • (a)

    η(y, y + tη(x, y)) = −tη(x, y),

  • (b)

    η(x, y + tη(x, y)) = (1 − t)η(x, y);

  • (a)

    η(y, y + tη(x, y)) = −tη(x, y),

  • (b)

    η(x, y + tη(x, y)) = (1 − t)η(x, y);

• (ii)

  K is an invex set with respect to η;

• (iii)

  for each x ≠ y, some α₀ ∈ [0,1],

  • (a)

    F_*(y)(α₀) > F_*(x)(α₀) implies $$,

•  

  or

  • (b)

    F^*(y)(α₀) > F^*(x)(α₀) implies $$;

• (iv)

  ∇F : K → (E) ⁿ is fuzzy strictly pseudo-invex monotone with respect to η on k.

Then, F is a fuzzy strictly pseudoinvex mapping η on k.

Proof. Let x, y ∈ K, x ≠ y, such that η(x, y) ^T∇F(y)⪰0. Thus, we need to show that F(x)≻F(y).

By contradiction, suppose that $$; then, there exists some α₀ ∈ [0,1], such that

or Without loss of generality, suppose that By hypothesis (iii), for some $$.

It follows from (i) and above inequity (105) that

Since ∇F is a strictly pseudo-invex monotone with respect to η, thus From $$ and $$, it follows that This contradicts η(x, y) ^T∇F^*(y)(α) ≥ 0.

Definition 20. Let K be an invex set with respect to η(x, x^*) : K × K → Rⁿ. A fuzzy mapping F : K → (E) ⁿ is called η-hemicontinuous, if for x, y ∈ K, for all α ∈ [0,1], the mappings t → F^*(y + tη(x, y))(α) and t → F_*(y + tη(x, y))(α) are continuous at 0⁺, with t ∈ [0,1].

Lemma 21. Let K be a nonempty convex set in Rⁿ, and suppose that

• (i)

  F : K → (E) ⁿ is a fuzzy pseudo-invex monotone with respect to η and η-hemicontinuous on K;

• (ii)

  η : K × K → Rⁿ satisfies

  • (a)

    η(y, y + λη(x, y)) = −λη(x, y), for all x, y ∈ K, λ ∈ [0,1],

  • (b)

    η(x, y + λη(x, y)) = (1 − λ)η(x, y), for all x, y ∈ K, λ ∈ [0,1];

• (iii)

  for any fixed y ∈ K, x → η(x, y) is linear; that is, for x⁽ⁱ⁾ ∈ K, i = 1,2, …, n, t_i ∈ [0,1], $$, with $$.

Then, for y ∈ K, $$, for all x ∈ K if and only if $$, for all x ∈ K.

Proof. ⇒ By contradiction, suppose that there exists a $$, such that $$. Thus, there exists some α₀ ∈ [0,1],

or Without loss of generality, assume that Since F : K → (E) ⁿ is a fuzzy pseudo-invex monotone with respect to η, thus That is, there are for all α ∈ [0,1].

Therefore,

In particular, for α₀ ∈ [0,1], This contradicts $$.

⇐ By contradiction, suppose that there exists a $$, such that $$. Thus, there exists some α₀ ∈ [0,1],

or Without loss of generality, assume that Since K is an invex set and by condition (a), we knowthat When η satisfies (iii), there is Since F is η-hemicontinuous on K, it follows that Therefore, Let $$, then This contradicts $$, for all x ∈ K.

By the hypothesis of the pseudo-invex monotonicity of F and the linearity of η, the existence theorem can be obtained.

Theorem 22. Let K be a nonempty convex set in Rⁿ, suppose that

• (i)

  F : K → (E) ⁿ is fuzzy pseudo-invex monotone with respect to η and η-hemicontinuous on K;

• (ii)

  η : K × K → Rⁿ satisfies

  • (a)

    η(y, y + λη(x, y)) = −λη(x, y), for all x, y ∈ K, λ ∈ [0,1],

  • (b)

    η(x, y + λη(x, y)) = (1 − λ)η(x, y), for all x, y ∈ K, λ ∈ [0,1];

• (iii)

  for any fixed y ∈ K, x → η(x, y) is linear; that is, for x⁽ⁱ⁾ ∈ K, i = 1,2, …, n, t_i ∈ [0,1], i = 1,2, …, n, for $$, with $$.

Then, there exists x^* ∈ K, such that $$, for all x ∈ K.

Proof. Let

That is, for all α ∈ [0,1], there are Let x⁽ⁱ⁾ ∈ K, i = 1,2, …, n, t_i ∈ [0,1], $$. Suppose that Then, there exists some α₀ ∈ [0,1], such that Therefore, From (iii), for fixed x ∈ K, That is, From η(x, x) = 0, it follows that 0 < 0, which is absurd. So $$.

Similarly, $$ can be proofed. Thus,

can be obtained. Therefore, Y^*(y)(α), Y_*(y)(α) are KKM mappings.

Let

That is, for all α ∈ [0,1], Let x ∈ Y(y); that is, $$. By Lemma 21, $$ holds, that is, $$. Thus, As Y^*(y)(α), Y_*(y)(α) are KKM mappings, thus $$, $$ are also KKM mappings. By Lemma 21, we have As $$, $$ are closed for every y ∈ K and K is a bounded set, then $$, $$ are bounded; hence, $$, $$ are compact. Therefore, Hence, there exists x^* ∈ K, such that That is,

The pseudo-invex monotonicity of F assures us of the existence of a solution to (FVLI), but not the uniqueness of such a solution. To achieve this, we assume the strictly pseudo-invex monotonicity of F.

Theorem 23. Let K be a nonempty convex set in Rⁿ, and suppose that

• (i)

  F : K → (E) ⁿ is strictly fuzzy pseudo-invex monotone with respect to η and η-hemicontinuous on K;

• (ii)

  η : K × K → Rⁿ is a skew function and satisfies

  • (a)

    η(y, y + λη(x, y)) = −λη(x, y), for all x, y ∈ K, λ ∈ [0,1],

  • (b)

    η(x, y + λη(x, y)) = (1 − λ)η(x, y), for all x, y ∈ K, λ ∈ [0,1];

• (iii)

  for any fixed y ∈ K, x → η(x, y) is linear; that is, for x⁽ⁱ⁾ ∈ K, i = 1,2, …, n, t_i ∈ [0,1], $$, with $$.

Then, there exists unique x^* ∈ K, such that $$, for all x ∈ K.

Proof. From Theorem 17, F is a fuzzy pseudo-invex monotone. Also from Theorem 22, there exists a solution for problem (FVLI).

Suppose that (FVLI) has two distinct solutions x^*, $$. Then,

Since F is a fuzzy strictly pseudo-invex monotone on K, then That is, there are for all α ∈ [0,1].

Since η is a skew function, there is $$. Thus

That is, which contradicts $$.

Lemma 24 (see Wu and Xu [6].)Let F : K → E be a fuzzy differentiable pseudoinvex mapping. If (x^*, ∇F(x^*)) is a solution of (FVLI), then x^* is a local optimal solution of (FP).

Lemma 25 (see Wu and Xu [6].)Let F : K → E be a fuzzy differentiable strictly pseudoinvex mapping. If (x^*, ∇F(x^*)) is a solution of (FVLI), then x^* is a strictly local optimal solution of (FP).

Theorem 26. Let F : K → E be a differentiable fuzzy mapping, and suppose that

• (i)

  η satisfies the following conditions:

  • (a)

    η(y, y + tη(x, y)) = −tη(x, y),

  • (b)

    η(x, y + tη(x, y)) = (1 − t)η(x, y);

• (ii)

  K is an invex set with respect to η;

• (iii)

  for each x ≠ y, some α₀ ∈ [0,1],

  • (a)

    F_*(y)(α₀) > F_*(x)(α₀) implies $$,

•  

  or

  • (b)

    F^*(y)(α₀) > F^*(x)(α₀) implies $$;

• (iv)

  ∇F : K → (E) ⁿ is fuzzy pseudo-invex monotone with respect to η on k.

If (x^*, ∇F(x^*)) is a solution of (FVLI), then x^* is a local optimal solution of (FP).

Proof. From Theorem 18, we know that F is a fuzzy pseudo invex. By Lemma 24, we can show it.

Theorem 27. Let F : K → E be a differentiable fuzzy mapping, and suppose that

• (i)

  η satisfies the following conditions:

  • (a)

    η(y, y + tη(x, y)) = −tη(x, y),

  • (b)

    η(x, y + tη(x, y)) = (1 − t)η(x, y);

• (ii)

  K is an invex set with respect to η;

• (iii)

  for each x ≠ y, some α₀ ∈ [0,1],

  • (a)

    F_*(y)(α₀) > F_*(x)(α₀) implies  $$,

•  

  or

  • (b)

    F^*(y)(α₀) > F^*(x)(α₀) implies $$;

• (iv)

  ∇F : K → (E) ⁿ is fuzzy strictly pseudo-invex monotone with respect to η on k.

If (x^*, ∇F(x^*)) is a solution of (FVLI), then x^* is a strictly local optimal solution of (FP).

Proof. From Theorem 19, we know that F is a fuzzy strictly pseudo-invex. By Lemma 25, we can show it.

Theorem 28. Let K be an invex set with respect to η : K  ×  K → Rⁿ. Suppose that

• (i)

  F : K → (E) ⁿ is fuzzy pseudo-invex monotone with respect to η and η-hemicontinuous on K;

• (ii)

  η satisfies the following conditions:

  • (a)

    η(y, y + λη(x, y)) = −λη(x, y),

  • (b)

    η(x, y + λη(x, y)) = (1 − λ)η(x, y).

Then, x^* ∈ K satisfies $$, for all x ∈ K if and only if it satisfies $$, for all x ∈ K.

Proof. ⇒ By contradiction, suppose that there exists an $$, such that $$. Thus, there exists some α₀ ∈ [0,1],

or Without loss of generality, assume that Since F : K → (E) ⁿ is fuzzy pseudo-invex monotone with respect to η, thus That is, for any α ∈ [0,1], Therefore, In particularly, for α₀ ∈ [0,1], there is This contradicts $$.

⇐ By contradiction, suppose that there exists an $$, such that $$. Thus, there exists some α₀ ∈ [0,1],

or Without loss of generality, assume that Since K is invex set with respect to η and assumption (ii), it follows that Again, since F is η-hemicontinuous on K, there is Therefore, Set $$, then This contradicts $$, for all x ∈ K.

Theorem 29. The fuzzy variational-like inequality (FVLI) is equivalent to the fuzzy generalized complementarity problem (FCP), when K is a convex cone.

Proof. At first, we show (FCP) ⊂ (FVLI). Suppose that (x^*, F(x^*)) is a solution of (FCP), then

Combining (158) and (159), we have Therefore, (x^*, F(x^*)) is also a solution of (FVLI).

Next, we show (FVLI) ⊂ (FCP). Let (x^*, F(x^*)) be a solution of (FVLI) which the degree of membership α ∈ [0,1], then

where $$ is the tolerance level which a decision maker can tolerate in the accomplishment of the fuzzy variation-like inequality $$. By contradiction, suppose that F(x^*) ^Tx^* ≠ 0, then there exists some α₀ ∈ [0,1], $$, such that or Or there exists some α₁ ∈ [0,1], $$, such that or Without loss of generality, suppose that Since K is a convex cone, then when y = λx^* with λ > 1, there is When y = 0, If $$, then This leads to a contradiction.

If $$, then

This also leads to a contradiction. Therefore, F(x^*) ^Tx^* = 0. Furthermore, from (161), it follows that Therefore, This shows that (x^*, F(x^*)) is a solution of (FCP).

Step 1. Transforming (FVLI) into problem (175).

Step 2. Initialize.

• (i)

  Input an acceptable satisfaction degree α₀ and the largest max-gen and pop-size.

• (ii)

  Input the criteria set: C_I = 1,2, …, n, n + 1, …, n + 2m + 2, n + 2m + 3, where j = 1,2, …, n stand for the variables, j = n + 1, …, n + 2m + 2 stand for inequality constraints to (175), and j = n + 2m + 3 stands for the sum of weighted satisfaction degree, respectively. Give the initial values and the upper and the lower values of criteria r, r_i ∈ C_I.

• (iii)

  Input the types of membership function that describes the fuzzy constraints.

• (iv)

  Input the weights w_j.

Step 3. Randomly produce the initial population and calculate their membership degrees by $$, where ξ ∈ U(0,1), i = 1,2, …, n; j = 1,2, …, pop-size and $$ is the upper bound of the ith element of variable x. The membership function $$ is calculated with (182).

Step 4. Set iteration index k = 1.

Step 5. Calculate the fitness function F(j) and selection probabilities P(j) by means of (183).

Step 6. Produce new individual x^(j) with the parent x⁽ⁱ⁾ as x^(j)k = x^(i)(k−1) + β^kG(x^(i)(k−1)). G(x) is defined by (179).

Step 7. For individual j, calculate the membership function $$ with (182), and update optimal degree of membership μ_max and the upper and the lower values of criteria r.

Step 8. Set k + 1 = k; if k ≤ max-gen, go to Step 5; otherwise, go to Step 9.

Step 9. Output the optimal membership degree μ_max and the upper and the lower values of criteria preferred by the decision maker, then stop.